The Games and Puzzles Journal — Issue 35, SeptemberOctober 2004 
Editor's Note: This article has been delayed in publication mainly because I was hoping to find a better way of presenting the dissection diagrams. Mr Willcocks provided A4size diagrams prepared for him by Dr J. D. Skinner, but it was not possible to reproduce them adequately on the screen. However, George Bell (19 December 2004) has now been able to provide a scale drawing gifimage of the first dissection (A1) as shown above. The impossibility of showing an accurate diagram is evident from the fact that the square side is 6732 units and the screen width is only 800 pixels. So the above image, which is 600 pixels wide, compresses 11.22 units to a pixel. So squares smaller than this, as occur in B1, B3, C1, C2 below, would be lost. For the diagrams in the text I have used the conventional method of showing the squares as rectangles.
During the past 75 years much work has been done on the problem of dividing a rectangle into smaller squares all of different sizes. When the rectangle itself is a square the problem is known as Squaring the Square and a comprehensive theory was developed by four mathematicians at Cambridge by which they showed a relationship between the elements of dissections and the currents in certain electrical networks. Details may be found in [1] and [2]. A very interesting nonmathematical but authoritative account may be found in [3]. The following notes are based entirely on the electrical theory.
In the diagram of Fig.(1a) ABC represents a network where for simplicity interior wiring has been omitted. DEF represents the map which is the dual of an equivalent map to ABC (interior wiring again omitted). These two networks are joined by the wires BE, BD, DC, CF, and FA. If now a current enters the network at A and leaves at E, the theory tells us that (1) the network gives rise to a squared square and (2) the wires BD and CF carry zero currents.
Lying in my bath one evening with a mental picture of such a network, I realised that the wire XD could equally well be drawn as XB and similarly the wire YB could be drawn as YD. What I had done was to replace two sides of the quadrilateral BYDX with the other two sides. It seemed that the natural thing to do was to replace the diagonal BD with the other diagonal XY and see what resulted. I had no reason to expect any interesting outcome, but on evaluating the new network I discovered that (1) I had a squared square, (2) the currents in BX and DY were numerically equal, and (3) vectors could be assigned as shown in Fig.(1b). Knowledge of (2) and (3) can simplify the calculation of currents in the network.
Experiment showed that a similar procedure could be followed with the wire CF or with both wires with a view to obtaining a single pair of equal squares in two different ways or two pairs of equal squares in one way. I have constructed a considerable number of such squares without finding a failure, but I lack a strict proof justifying the procedure. As with all other methods of this kind it is important to check that there is no ‘unwanted’ imperfection.
An example of three squares constructed in this way showing an interesting arithmetical relationship is given in the accompanying diagrams C1, C2, C3. The sum of the duplicated squares in C1 and C2 (92 + 208) is equal to the sum of the two duplicated pairs in C3 (97 + 203).
This method is not restricted to the use of selfdual maps. A dissection of a rectangle in two different ways but with a corner square in one equal to a corner square in the other will yield two different tripolar networks and one of these can be used with the dual of the other.
In this way we can form four networks, and it is their close relationships with each other which result in some curious arithmetical equalities, such as that noted above. From A and B we have 431 + 187 = 332 + 286.
The writer does not claim that all initial networks give such interesting results. He has been content to show some that he has found.
A1: 34 squares, side 6732 units.
3006  1870  1856  
651  1205  
1136  734  
97  554  
831  
1456  865  685  513  623  
374  1385  
194  1011  
403  110  
927  
454  231  
591  274  
634  
317  411  
110  2286  
2270  94  
2176 
A2: 34 squares, side 6876 units.
2637  2305  1934  
714  1220  
332  1422  551  
1834  1135  
208  506  
759  
461  1265  
416  804  
481  609  332  
699  436  
28  388  
360  
263  654  
526  83  
433  2457  
2405  391  
2014 
A3: 35 squares, side 9447 units.
3561  3130  2756  
1002  1754  
431  1821  878  
2476  1516  
250  752  
1128  
626  1880  
682  708  431  
960  556  
500  1254  
177  254  
100  77  
404  834  
831  
808  
3410  430  
77  3057  
2980 
B1: 34 squares, side 6705 units.
2592  1681  2432  
911  770  
202  2230  
141  446  183  
1285  1307  747  305  
385  
629  122  
507  
560  187  
1323  
1263  22  
939  950  
187  2043  
302  626  11  
615  1856  
1565  
1241 
B2: 34 squares, side 6732 units.
2432  2293  2007  
286  1721  
842  1737  
1870  562  
421  141  
374  669  
327  94  
233  235  
302  1419  
560  
558  286  
272  1102  751  
2430  830  
51  634  166  
1585  
1600  483  
1117 
B3: 35 squares, side 4669 units.
1804  1526  1339  
187  1152  
604  1109  
1490  314  
302  12  
298  318  
294  8  
286  20  
151  187  
144  1008  
115  36  
774  702  
695  
1375  115  
72  468  162  
1260  396  
1170  
864 
C1: 34 squares, side 6705 units.
2817  2149  1739  
410  1329  
668  1048  843  
1555  777  485  288  380  
205  562  76  
1405  
196  92  
1368  357  
1  195  
292  194  
389  
919  
778  291  
680  
2333  
92  2232  
2140 
C2: 34 squares, side 6876 units.
2637  1762  2477  
875  887  
706  1771  
1175  644  818  513  362  
353  534  
359  3  
356  
305  208  
175  1065  
470  174  
1098  
296  1001  
61  705  
1236  
208  2628  
592  2420  
1828 
C3: 35 squares, side 9413 units.
3509  3306  2598  
708  1890  
203  1153  1506  1152  
2404  1308  
354  768  30  
358  380  212  203  
1920  
53  150  
168  44  
97  
1693  414  
1096  570  
548  
1182  
1118  
3508  
97  3005  
2908 
It is possible to make further speculations. It is obvious that the process is reversible. The method employed results in all cases with a square containing 3 component squares situated as in Fig.(2a). Reversing the process the two squares A are removed and a zero current is introduced.
Fig.2a

Fig.2b

What happens if we remove by this process in a square with a configuration as in Fig.(2b) the two squares A which adjoin square D? The small number of examples I have tested have all yielded squared squares.
The sceptical reader might care to test the inverse method with the following square, Fig.(3). At first sight this does not look like a candidate but it can be made one by treating the square [4] as being 4 squares of side 2 surrounding a zero square. To my surprise I found that this resulted in a square (imperfect of side 313). Is it conceivable that a perfect square could be found in this way?
Fig.(3). 21 squares, side 107
34  36  37  
32  2  
21  17  
16  21  
4  20  9  
9  16  
4  17  
13  
41  
34  2  
32 
In conclusion I would like to express my thanks to Dr J. D. Skinner for his help with the diagrams.
References.
[1] The dissection of rectangles into squares R. L. Brooks, C. A. B. Smith, A. H. Stone, W. T. Tutte, (Duke Math J. Vol.7 p.312340)
[2] Squaring the Square W. T. Tutte (Can. J. Math. Vol.2 p.197209, 1950).
[3] Squaring the Square W. T. Tutte (Chapter 17 of the 2nd Scientific American Book of Mathematical Puzzles and Diversions).
George Bell cites the following website on squaring the square, though I've not been able to access it properly since its "javascript" does not pass my computer's security settings. www.squaring.net
Update (2 June 2013): My more recent computers have had no trouble with access to the site. I have corresponded with the author Stuart Anderson. There is now a page devoted to T. H. Willcocks and his work: THW on www.squaring.net. Mr Willcocks reached the age of 100 on 19 April 2012. 